Why Is Capacitance Considered Constant in a Conductor?

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    Capacitance Conductor
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Discussion Overview

The discussion revolves around the concept of capacitance in conductors, specifically questioning why capacitance is considered a constant despite the non-linear relationship between charge and potential. Participants explore theoretical implications and mathematical relationships related to capacitance, charge, and potential.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the constancy of capacitance, arguing that the potential of a conductor is not a linear function of the charge added, suggesting that more charge requires more work to add.
  • Another participant clarifies that potential is linear with respect to the total charge on the conductor, not the incremental charge added, emphasizing that ##V## is proportional to ##Q##.
  • A participant asserts that while capacitance is a constant ratio of charge to potential (C=Q/V), the relationship between charge and voltage does not need to be linear as long as their ratio remains constant.
  • Some participants discuss the implications of charging and discharging capacitors under different conditions, noting that the physical properties of the capacitor determine capacitance, which remains constant unless those properties change.
  • Quantitative reasoning is introduced, with a participant stating that the work done to add charge depends on the existing charge, but another counters that the change in potential with respect to charge is constant, represented as ##dV/dQ = 1/C##.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the relationship between charge and potential, with some asserting that capacitance is constant while others challenge the linearity of the relationship. The discussion remains unresolved regarding the implications of these relationships.

Contextual Notes

Participants reference the physical properties of capacitors and their dependence on dimensions and materials, but there is no consensus on how these properties interact with the charge and potential relationships in conductors.

Higgsono
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The capacitance C of a conductor is given to be a constant relationship between charge Q and potential V of the conductor given by Q = CV.
But how can C be a constant? Because the potential of the conductor will not be a linear relationship of the charge that I add. THe more charge there is on the conductor already, the more work is needed to add additional charge. Hence the potential I add by bringing new charges to the conductor must depend on the charge already there.

What is it that I don't understand?
 
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Higgsono said:
Because the potential of the conductor will not be a linear relationship of the charge that I add.
The potential is linear with the charge that is on the conductor, not the charge that is added. I.e. ##V## is proportional to ##Q## not ##\Delta Q##
 
Dale said:
The potential is linear with the charge that is on the conductor, not the charge that is added. I.e. ##V## is proportional to ##Q## not ##\Delta Q##

huh? Q and V are not constants. I must be able to double the charge and the relation should be the same right?
 
Higgsono said:
huh? Q and V are not constants. I must be able to double the charge and the relation should be the same right?
Yes, the proportionality between ##Q## and ##V## is constant, but what you are describing in your text is the relationship between ##\Delta Q## and ##\Delta W##. ##\Delta Q\ne Q## and ##\Delta W \ne V##
 
Remember that C is a physical quantity. Remember back to the simple capacitance days and how two plates make a capacitor. The capacitance depends on dimensions and material properties. It does not depend on electrical properties. It is a constant as long as the materials and the physical dimensions don't change.

As for the charge and voltage, capacitance is the ratio of the two, C=Q/V; therefore Q and V don't have to be linear, they can follow any line as long as the ratio between them remains constant.

If you want to look at the different lines that Q and V follow, look at charging and discharging a capacitor at constant current versus constant voltage/resistance.
 
Let’s try it more quantitatively
Higgsono said:
Because the potential of the conductor will not be a linear relationship of the charge that I add.
That is correct. ##V \propto Q## not ##V \propto dQ##

Higgsono said:
THe more charge there is on the conductor already, the more work is needed to add additional charge.
Yes, ##dW/dQ=f(Q)## where ##df/dQ>0##

Specifically
##dW/dt=P=IV=V \; dQ/dt##
##dW/dQ = V = Q/C =f(Q)##
So ##df/dQ =1/C>0##

Higgsono said:
Hence the potential I add by bringing new charges to the conductor must depend on the charge already there.
No, it does not follow. Instead ##dV/dQ = 1/C##
 

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